The Central Idea
Measures of center and position reduce a dataset to a few informative numbers.
The most common introductory summaries are:
- mean — an arithmetic balance point;
- median — the middle ordered value;
- mode — the most frequent value or category;
- quantiles — values that divide ordered data into specified proportions.
These summaries are related, but they are not interchangeable.
The Arithmetic Mean
The arithmetic mean is found by adding all observed values and dividing by the number of observations.
For sample values
\[ x_1,x_2,\ldots,x_n, \]the sample mean is
\[ \boxed{ \bar{x} = \frac{x_1+x_2+\cdots+x_n}{n} }. \]Using summation notation,
\[ \boxed{ \bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i }. \]Example: Computing the Mean
Consider
\[ 4,\ 6,\ 7,\ 8,\ 10. \]Their sum is
\[ 4+6+7+8+10=35. \]There are \(5\) observations, so
\[ \bar{x} = \frac{35}{5} = \boxed{7}. \]The Mean as a Balance Point
The mean has an important balancing property.
If deviations from the mean are defined by
\[ x_i-\bar{x}, \]then
\[ \boxed{ \sum_{i=1}^{n}(x_i-\bar{x})=0 }. \]Positive deviations above the mean balance negative deviations below it.
Example: Deviations from the Mean
For
\[ 4,\ 6,\ 7,\ 8,\ 10, \]the mean is \(7\). The deviations are
\[ -3,\ -1,\ 0,\ 1,\ 3. \]Their sum is
\[ -3-1+0+1+3=0. \]The Population Mean
If the values describe an entire population of size \(N\), the population mean is commonly written
\[ \boxed{ \mu = \frac{1}{N}\sum_{i=1}^{N}x_i }. \]The symbol \(\bar{x}\) commonly denotes a sample mean, while \(\mu\) commonly denotes a population mean.
Weighted Mean
Sometimes observations contribute with different weights.
If values \(x_1,\ldots,x_k\) have weights \(w_1,\ldots,w_k\), then the weighted mean is
\[ \boxed{ \bar{x}_w = \frac{\sum_{i=1}^{k}w_i x_i} {\sum_{i=1}^{k}w_i} }. \]Example: Weighted Mean
Suppose a course grade uses:
- homework: \(80\) with weight \(20\%\);
- midterm: \(90\) with weight \(30\%\);
- final: \(85\) with weight \(50\%\).
Mean from a Frequency Table
If a value \(x_i\) occurs with frequency \(f_i\), then
\[ \boxed{ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} }. \]The frequencies act as weights.
Example: Mean from Frequencies
| Value \(x\) | Frequency \(f\) | \(fx\) |
|---|---|---|
| 1 | 2 | 2 |
| 2 | 3 | 6 |
| 3 | 1 | 3 |
Therefore,
\[ \bar{x} = \frac{2+6+3}{2+3+1} = \frac{11}{6} \approx \boxed{1.83}. \]The Median
The median is the middle value of an ordered dataset.
The first step is always to arrange the observations from smallest to largest.
Median with an Odd Number of Observations
Consider
\[ 3,\ 5,\ 8,\ 11,\ 14. \]There are \(5\) ordered observations.
The middle observation is the third:
\[ \boxed{\operatorname{median}=8}. \]Median with an Even Number of Observations
Consider
\[ 3,\ 5,\ 8,\ 11,\ 14,\ 17. \]The two middle values are \(8\) and \(11\).
The median is their average:
\[ \operatorname{median} = \frac{8+11}{2} = \boxed{9.5}. \]Median Position
For an ordered dataset of size \(n\), the conceptual middle position is
\[ \frac{n+1}{2}. \]If this is an integer, it identifies the single middle observation.
If it falls between two positions, the conventional sample median averages the two middle values.
The Median Divides Ordered Data
The median is a positional measure.
Roughly half of the observations lie at or below the median and roughly half lie at or above it.
Repeated values can make the exact percentages on each side differ from \(50\%\), but the median still identifies the center of the ordered data.
The Mode
The mode is the value or category that occurs most often.
For
\[ 2,\ 3,\ 3,\ 3,\ 5,\ 7, \]the mode is
\[ \boxed{3}. \]The Mode Can Be Categorical
Unlike the mean and median, the mode can be used with nominal categorical data.
Suppose payment methods occur with frequencies:
| Method | Frequency |
|---|---|
| Cash | 18 |
| Credit | 31 |
| Debit | 24 |
Credit is the modal category.
More Than One Mode
A dataset may have:
- one mode;
- two modes;
- several modes;
- no unique mode.
For
\[ 1,\ 1,\ 2,\ 2,\ 3, \]both \(1\) and \(2\) occur most often.
The dataset is bimodal.
Mean, Median, and Mode Answer Different Questions
| Measure | Main Question |
|---|---|
| Mean | What is the arithmetic balance point? |
| Median | What value lies in the middle of the ordered data? |
| Mode | What value or category occurs most often? |
Effect of an Outlier on the Mean
Consider
\[ 4,\ 5,\ 5,\ 6,\ 7. \]The mean is
\[ \frac{4+5+5+6+7}{5} = \frac{27}{5} = 5.4. \]Now replace \(7\) with \(70\):
\[ 4,\ 5,\ 5,\ 6,\ 70. \]The new mean is
\[ \frac{90}{5} = 18. \]One extreme value moved the mean substantially.
Effect of an Outlier on the Median
For both
\[ 4,\ 5,\ 5,\ 6,\ 7 \]and
\[ 4,\ 5,\ 5,\ 6,\ 70, \]the median is
\[ \boxed{5}. \]The median depends mainly on ordered position, so it is much less sensitive to a single extreme value.
Resistance to Extreme Values
A statistic is called resistant if unusual or extreme observations do not change it greatly.
The median is relatively resistant.
The mean is not resistant because every numerical value contributes directly to its calculation.
Center in Symmetric Distributions
In a roughly symmetric, unimodal distribution, the mean and median are often close together.
If the distribution is also strongly concentrated around one central value, the mode may lie near them as well.
Center in Right-Skewed Distributions
In a strongly right-skewed distribution, unusually large values tend to pull the mean toward the right tail.
A common pattern is
\[ \text{mode} < \text{median} < \text{mean}, \]although this ordering is not a universal law for every dataset.
Center in Left-Skewed Distributions
In a strongly left-skewed distribution, unusually small values tend to pull the mean toward the left tail.
A common pattern is
\[ \text{mean} < \text{median} < \text{mode}, \]again as a useful tendency rather than an absolute rule.
What Is a Quantile?
A quantile is a value that divides ordered data according to a specified cumulative proportion.
Examples include:
- quartiles;
- deciles;
- percentiles.
Quantiles describe relative position within a distribution.
Quartiles
Quartiles divide ordered data into four parts.
- \(Q_1\): first quartile;
- \(Q_2\): second quartile;
- \(Q_3\): third quartile.
The second quartile is the median:
\[ \boxed{Q_2=\operatorname{median}}. \]Interpretation of Quartiles
Approximately:
- \(25\%\) of observations are at or below \(Q_1\);
- \(50\%\) are at or below \(Q_2\);
- \(75\%\) are at or below \(Q_3\).
Ties and finite-sample conventions can affect exact counts, but the cumulative interpretation remains the central idea.
Example: Finding Quartiles
Consider the ordered dataset
\[ 2,\ 4,\ 5,\ 7,\ 8,\ 10,\ 12,\ 13,\ 15. \]The median is
\[ Q_2=8. \]Using the common convention of excluding the median when splitting an odd dataset, the lower half is
\[ 2,\ 4,\ 5,\ 7 \]and the upper half is
\[ 10,\ 12,\ 13,\ 15. \]Therefore,
\[ Q_1 = \frac{4+5}{2} = 4.5, \] \[ Q_3 = \frac{12+13}{2} = 12.5. \]Quartile Conventions Can Differ
Different textbooks and software packages use slightly different rules for quartiles and percentiles, especially for small datasets.
Two correct systems can therefore produce slightly different numerical quartiles.
When precision matters, state the convention or software method being used.
Percentiles
A percentile indicates a position in an ordered distribution based on percentage.
The \(p\)th percentile is a value associated with approximately \(p\%\) of the observations lying at or below it.
For example, the \(90\)th percentile marks a high position in the distribution.
Percentile Rank
A percentile rank describes the percentage of observations at or below a particular value, subject to the convention being used.
If a score is at the \(80\)th percentile, it is positioned at or above approximately \(80\%\) of the reference observations.
It does not mean that the person scored \(80\%\) of the possible points.
Deciles
Deciles divide ordered data into ten parts.
The first decile corresponds roughly to the \(10\)th percentile, the second to the \(20\)th percentile, and so on.
Quantile Notation
A general quantile can be described by a probability level \(p\), where
\[ 0\le p\le1. \]A \(p\)-quantile identifies a value associated with cumulative proportion \(p\).
Thus,
\[ p=0.25 \]corresponds to the first quartile, while
\[ p=0.50 \]corresponds to the median.
Five-Number Summary
The five-number summary consists of:
\[ \boxed{ \text{minimum},\quad Q_1,\quad \text{median},\quad Q_3,\quad \text{maximum} }. \]These five values provide a compact description of ordered position and spread.
Example: Five-Number Summary
For
\[ 2,\ 4,\ 5,\ 7,\ 8,\ 10,\ 12,\ 13,\ 15, \]using the quartile convention above:
\[ \boxed{ 2,\ 4.5,\ 8,\ 12.5,\ 15 }. \]Choosing Between Mean and Median
The mean is often useful when:
- all numerical values should influence the center;
- the distribution is roughly symmetric;
- later calculations depend algebraically on the mean.
The median is often useful when:
- the distribution is strongly skewed;
- extreme values are present;
- ordered position is more important than arithmetic balance.
Example: Typical Income
Suppose nine employees earn amounts near \(\$50{,}000\), while one executive earns \(\$1{,}000{,}000\).
The mean salary will be pulled upward substantially.
The median may provide a more representative description of a typical employee's salary.
Example: Total Resource Allocation
Suppose a company wants to calculate total payroll from average salary.
The mean is directly connected to the total:
\[ \boxed{ \text{total} = n\bar{x} }. \]In that context, the mean may be operationally more useful than the median.
Common Error: Computing the Median Before Sorting
Consider
\[ 9,\ 2,\ 7,\ 4,\ 5. \]The middle entry as written is \(7\), but the ordered data are
\[ 2,\ 4,\ 5,\ 7,\ 9. \]Therefore,
\[ \boxed{\operatorname{median}=5}. \]Common Error: Dividing by the Wrong Number
The mean divides by the number of observations, not by the number of distinct values.
For
\[ 2,\ 2,\ 2,\ 8, \]the mean is
\[ \frac{2+2+2+8}{4} = \frac{14}{4} = 3.5. \]Common Error: Averaging Categories
Numerical codes for nominal categories should not generally be averaged.
If
\[ 1=\text{cash},\quad 2=\text{credit},\quad 3=\text{debit}, \]then a mean code of \(2.1\) does not describe a meaningful average payment method.
Common Error: Assuming the Mode Is Unique
A dataset may have several values tied for highest frequency.
Do not force a single mode when the frequency structure does not support one.
Common Error: Confusing Percentile with Percentage Score
A student who scores \(75\%\) on a test has earned \(75\%\) of the available points.
A student at the \(75\)th percentile is positioned relative to other scores.
These are different concepts.
Common Error: Ignoring Outliers When Reporting the Mean
A mean can be mathematically correct but still provide a poor description of a typical observation when the distribution is highly skewed or contains extreme values.
Always inspect the distribution before choosing a measure of center.
A Reliable Procedure for Measures of Center
- Identify the variable and confirm that the proposed summary is meaningful.
- Inspect the data or display for skewness, gaps, and unusual values.
- Compute the mean when arithmetic balance is relevant.
- Order the observations before computing the median or quantiles.
- Use frequencies to identify the mode.
- Compare mean and median when distribution shape matters.
- Interpret each statistic in the units and context of the variable.
A Reliable Procedure for Quantiles
- Order the data from smallest to largest.
- Identify the desired cumulative proportion.
- Apply a stated quartile or percentile convention.
- Interpolate when the chosen convention requires it.
- Interpret the resulting value as a position within the ordered distribution.
- Use the same convention consistently when comparing datasets.
Practice 1: Mean
Find the mean of
\[ 5,\ 7,\ 8,\ 10,\ 10. \]Practice 2: Median
Find the median of
\[ 9,\ 3,\ 7,\ 2,\ 11. \]Practice 3: Even-Sized Median
Find the median of
\[ 4,\ 6,\ 8,\ 10,\ 12,\ 14. \]Practice 4: Mode
Find the mode or modes of
\[ 1,\ 2,\ 2,\ 3,\ 3,\ 4. \]Practice 5: Weighted Mean
A project score is \(90\) with weight \(40\%\), and an exam score is \(80\) with weight \(60\%\).
Find the weighted mean.
Practice 6: Mean from Frequencies
| Value | Frequency |
|---|---|
| 2 | 2 |
| 4 | 3 |
| 6 | 1 |
Find the mean.
Practice 7: Quartiles
Using the convention of excluding the median when splitting an odd-sized dataset, find \(Q_1\), \(Q_2\), and \(Q_3\) for
\[ 1,\ 3,\ 4,\ 6,\ 8,\ 9,\ 11,\ 14,\ 18. \]Practice 8: Mean or Median?
A neighborhood has nine homes valued between \(\$250{,}000\) and \(\$350{,}000\), plus one mansion valued at \(\$4{,}000{,}000\).
Which measure is likely to better describe a typical home value?
Practice 9: Percentile Interpretation
A test score is at the \(90\)th percentile.
Explain what this means.
Practice 10: Identify the Error
A student finds the median of
\[ 12,\ 4,\ 9,\ 7,\ 3 \]by selecting the middle number as written and reports \(9\).
Explain the error and find the correct median.
Practice Answers
Practice 1
\[ \bar{x} = \frac{5+7+8+10+10}{5} = \frac{40}{5} = \boxed{8}. \]Practice 2
Order the data:
\[ 2,\ 3,\ 7,\ 9,\ 11. \]Therefore,
\[ \boxed{\operatorname{median}=7}. \]Practice 3
\[ \operatorname{median} = \frac{8+10}{2} = \boxed{9}. \]Practice 4
Both \(2\) and \(3\) occur twice, more often than any other value.
\[ \boxed{\text{modes}=2,\ 3}. \]Practice 5
\[ 0.40(90)+0.60(80) = 36+48 = \boxed{84}. \]Practice 6
\[ \bar{x} = \frac{2(2)+3(4)+1(6)}{2+3+1} = \frac{22}{6} \approx \boxed{3.67}. \]Practice 7
The median is
\[ Q_2=8. \]Lower half:
\[ 1,\ 3,\ 4,\ 6. \] \[ Q_1 = \frac{3+4}{2} = 3.5. \]Upper half:
\[ 9,\ 11,\ 14,\ 18. \] \[ Q_3 = \frac{11+14}{2} = 12.5. \]Therefore,
\[ \boxed{Q_1=3.5,\quad Q_2=8,\quad Q_3=12.5}. \]Practice 8
The median is likely the better description of a typical home value because the mansion is an extreme high value that would pull the mean upward.
Practice 9
A score at the \(90\)th percentile is at or above approximately \(90\%\) of the scores in the reference distribution, subject to the percentile convention being used.
Practice 10
The data must first be ordered:
\[ 3,\ 4,\ 7,\ 9,\ 12. \]Therefore,
\[ \boxed{\operatorname{median}=7}. \]What to Remember
- The mean is the arithmetic average and balance point of quantitative data.
- The sample mean is \(\bar{x}=\frac{1}{n}\sum x_i\).
- The median is the middle value after the data are ordered.
- The mode is the most frequent value or category.
- The mean is sensitive to extreme observations.
- The median is relatively resistant to outliers and skewness.
- Quartiles divide ordered data into four parts.
- The second quartile is the median.
- Percentiles describe relative position within an ordered distribution.
- Different quantile conventions can produce slightly different results, so the method should be stated when precision matters.
Where We Go Next
Measures of center describe where observations are concentrated, but they do not tell us how widely the data are dispersed.
The next lesson, Range, Variance, Standard Deviation, and Interquartile Range, develops numerical measures of spread and shows how variability complements measures of center.